Ring theory

• When Do Finitely Generated Prime Ideals Imply a Noetherian Ring?
  For a commutative ring with identity, being Noetherian is equivalent to every prime ideal being finitely generated; here is the proof and key distinction.
• How to Prove x(y − z) = xy − xz in a Ring
  In any ring, left distributivity and the additive inverse law give x(y − z) = xy − xz; follow the proof and a worked matrix example.
• Why Is Every Finite Integral Domain a Field?
  In a finite integral domain, multiplication by any nonzero element is injective and therefore surjective, which guarantees a multiplicative inverse.
• What Are Examples of Infinite Boolean Rings?
  Examples of infinite Boolean rings include power sets under symmetric difference and intersection, finite-cofinite sets, and products of copies of F₂.
• Can an Artinian Ring Fail to Be Noetherian?
  Artinian commutative rings are Noetherian, but a right Artinian noncommutative ring can fail to be left Noetherian; here is an example.
• How to Make Addition and Multiplication Tables for a Ring or Ideal
  Build ring and ideal operation tables by applying the inherited operations to each pair; see a complete example using Z4 and its ideal {0, 2}.